Long cycles in certain graphs of large degree

Let G be a connected graph of order n and X={x∈V:d(x)≥n/2}. Suppose |X|≥3 and G satisfies the modified Fan's condition. We show that the vertices of the block B of G containing X form a cycle. This generalizes a result of Fan. We also give an efficient algorithm to obtain such a cycle. The complexity of this algorithm is O(n2). In case G is 2-connected, the condition |X|≥3 can be removed and G is hamiltonian.